Definition and Basic Properties of Distributions

Abstract

In the introduction we have seen how various difficulties in the theory of partial differential equations and in Fourier analysis lead one to extend the space of continuous functions to the space of distributions. In Section 2.1 we make the definition explicit and precise, using the properties of test functions proved in Chapter I. The weak topology in the space of distributions is also introduced there. The notion of support is extended to distributions in Section 2.2 and it is shown there that distributions may be defined locally provided that the local definitions are compatible. In addition it is proved that if u is a distribution then there is a unique way to define u(φ) for all φ∈C∞ with supp u ∩ supp φ compact. The problem of estimating u(φ) in terms of the derivatives of φ on the support of u only is discussed at some length in Section 2.3. The deepest result is Whitney’s extension theorem (Theorem 2.3.6). We shall rarely need the results which follow from it so the reader might prefer to skip the section from Theorem 2.3.6 on.